Computing the Matching Distance of 2-Parameter Persistence Modules from Critical Values

TL;DR

This work tackles the exact computation of the matching distance $d_{match}$ for 2-parameter persistence modules by decoupling the problem from dual-space methods and proving that, in the primal plane, the maximum bottleneck distance over lines of positive slope is attained on a finite set of lines determined by a finite set of critical values and switch points. The authors introduce Property (P) and (Q) to guarantee a finite, constructive search space and provide explicit formulas for switch points, including lines of diagonal slope, while proving that vertical/horizontal lines need not determine the maximum. They connect their approach to existing literature, show how line-equivalence classes can be refined to ensure consistent optimal matchings across lines, and present an algorithm with a quantified worst-case complexity that leverages parallelism. The results yield a practical, geometry-driven framework to compute $d_{match}$ exactly in dimension two, clarifying the roles of diagonal, vertical, horizontal, and switch-line directions and offering a path toward more efficient, finite-line computations in multi-parameter persistence. This advances the viability of multi-parameter persistence as a data-analysis tool by enabling exact, stable comparisons of 2D persistence modules.

Abstract

The exact computation of the matching distance for multi-parameter persistence modules is an active area of research in computational topology. Achieving an easily obtainable exact computation of this distance would permit multi-parameter persistent homology to be a viable option for data analysis. For this purpose, two approaches are currently available, limited to persistence with parameters from $\mathbb{R}^2$: authors of arXiv:1812.09085, arXiv:2111.10303 work in the discrete setting and apply the point-line duality; authors of arXiv:2210.16718, arXiv:2312.04201 work in the smooth setting while remaining in the primal plane. In this paper, we streamline the computation of the matching distance in the combinatorial setting while staying in the primal plane. In doing so, besides connecting results from the literature, we give explicit formulas for the switch points needed by all the available methods and we show that it is possible to avoid considering vertical and horizontal lines. For the latter, lines with slope 1 play an essential role.

Figures (8)

• Figure 1: The positive cone $S_+(u)$ of $u\in\mathbb{R}^2$ and the decomposition of its boundary into $S_{\{1\}}$, $S_{\{2\}}$ and $S_{\{1,2\}}$, which correspond respectively to the vertical boundary, horizontal boundary, and $u$.
• Figure 2: The push of $u$ along the line $L$. In this case, $A^L_u=\{2\}$, and so $u$ is to the left of $L$.
• Figure 3: The persistence module considered in \ref{['ex:need-diag']}: The matching distance is not attained at any line through two critical values.
• Figure 4: The persistence modules considered in \ref{['ex:diag-not-suff']}: The matching distance is not attained at any diagonal line.
• Figure 5: The persistence modules considered in \ref{['ex:need-omega']}: Neither lines through critical values nor diagonal lines suffice to achieve the matching distance.

Theorems & Definitions (44)

• Definition 2.1: Parameter Spaces
• Definition 2.2: Persistence Modules
• Definition 2.3: Interval Module
• Definition 2.4: Persistence Diagram
• Definition 2.5: Bottleneck Distance
• Remark 2.6
• Definition 2.7: Matching Distance
• Definition 2.8: Positive Cone
• Definition 2.9
• Lemma 2.10